1. ## Series converge?

Let $\displaystyle a_n>0$, $\displaystyle \lim_{n\to\infty}a_n=a>0$. Study the converegence or divergence of the series $\displaystyle \sum_{n=1}^\infty (a/a_n)^n$.

Thanks.

2. ## Re: Series converge?

Let $\displaystyle a_n>0$, $\displaystyle \lim_{n\to\infty}a_n=a>0$. Study the converegence or divergence of the series $\displaystyle \sum_{n=1}^\infty (a/a_n)^n$.

Thanks.
I think this series must diverge. If \displaystyle \displaystyle \begin{align*} a_n \to a \end{align*} then \displaystyle \displaystyle \begin{align*} \frac{a}{a_n} \to \frac{a}{a} = 1 \neq 0 \end{align*}.

3. ## Re: Series converge?

But we may have the limit $\displaystyle 1^\infty$.

4. ## Re: Series converge?

Originally Posted by Prove It
I think this series must diverge. If \displaystyle \displaystyle \begin{align*} a_n \to a \end{align*} then \displaystyle \displaystyle \begin{align*} \frac{a}{a_n} \to \frac{a}{a} = 1 \neq 0 \end{align*}.
For example,
$\displaystyle \lim_{n\to\infty}(1-1/\sqrt{n})^n=\lim_{n\to\infty}e^{-\sqrt{n}}=0.$